A Study of a Loss System with Priorities

TL;DR

This paper extends the Erlang loss formula to systems with preemptive priorities, deriving the result directly from global balance equations and analyzing the impact of service time distribution variance on blocking probabilities.

Contribution

It provides the first direct derivation of the preemptive priority loss system result from global balance equations and examines insensitivity properties in such systems.

Findings

Derived the preemptive priority loss system result from global balance equations.

Showed that blocking probabilities are sensitive to service time distribution variance.

Demonstrated that higher service time variance reduces blocking for lower priority customers.

Abstract

The Erlang loss formula, also known as the Erlang B formula, has been known for over a century and has been used in a wide range of applications, from telephony to hospital intensive care unit management. It provides the blocking probability of arriving customers to a loss system involving a finite number of servers without a waiting room. Because of the need to introduce priorities in many services, an extension of the Erlang B formula to the case of a loss system with preemptive priority is valuable and essential. This paper analytically establishes the consistency between the global balance (steady state) equations for a loss system with preemptive priorities and a known result obtained using traffic loss arguments for the same problem. This paper, for the first time, derives this known result directly from the global balance equations based on the relevant multidimensional Markov chain. The paper also addresses the question of whether or not the well-known insensitivity property of the Erlang loss system is also applicable to the case of a loss system with preemptive priorities, provides explanations, and demonstrates through simulations that, except for the blocking probability of the highest priority customers, the blocking probabilities of the other customers are sensitive to the holding time distributions and that a higher variance of the service time distribution leads to a lower blocking probability of the lower priority traffic.